I'm not sure, but isn't it the case that Bose-Einstein condensates have been created (a collection of super cooled particles) and shown to behave as if they were a single particle?
Specifically, haven't physicists achieved putting such a condensate in a superposition? Wouldn't that count as a larger scale demonstration of superposition?
I can see that a virus is going to be much more complex than a bunch of uniform particles, but why should scale matter whether superposition is possible? Don't all objects experience some level of superpostion, only it occurs for immeasurably small time periods before wave collapse?
One question that Schroedinger's Cat raises is "Is there a level of complexity that prevents a superposition forming?" and the proposed experiment will extend the upper limit of complexity that we know quantum mechanics applies to.
Exactly! But, isn't there a different (though related) question: Does it matter if the object (cat) is capable of making observations of its own? Is it possible that "being capable of observation" excludes the possibility of superposition?
I think we'd all agree that a virus isn't capable of making observations... but the cat is? (or if you have a problem with that, just place a person in the box)
Or maybe (BIG maybe), "being capable of observation" just requires a certain amount of complexity?
great theorems of mathematics. This book by William Dunham is a look at the lives and work of 10 or so great mathematicians throughout history.
The sections on Euclid and primes, Euler and infinite sums, and Cantor and the continuum are particularly good. Though it covers great and truly ingenious (even inspiring) mathematics, it should all be accessible to a high school student. Plus, the biographies are interesting in their own right and help to break up the parts where you actually have to concentrate.
I read this in my first year or so as an undergraduate, and I recommend it to anyone who wants to gain a feel for what mathematics is all about.
Exactly, its only the inclusion of gravity within the perspective of special relativity (which is called general relativity) that causes problems with quantum mechanical formulations.
On the contrary, advertising will flourish even more.
It seems that everyone can agree that there is a type of advertising which has as its main function to affect a person's subconscious tendencies toward a particular brand.
Now, if every brand employs this sort of advertising, to a more or less equal degree, then
exactly which brand does a person settle on? If the advertising is done right on each brand's
behalf, then statistically there will be roughly equal portions of the consumer base that will
prefer any particular brand. Any new brand that enters the fray will be at a gross disadvantage
unless they employ similar tactics. It will become economically impossible for them to sell their
brand name product without a comparative advertising campaign. Also, if any existing brand ups
its advertising effectiveness, then the other brands will have to follow suit, or risk losing
large portions of the consumer base. So you see, more and better advertising begets more and
better advertising, and a vicious cycle ensues.
More advertising agencies will emerge to handle the increasing volume of required advertising.
These agencies will compete amongst themselves (through advertising) to be employed by the
various brands. Existing agencies will need to increase their output further, too, and so
advertisers will begin to pay other advertisers to advertise their advertising!
Eventually, we will reach the advertising event horizon! -- where it becomes economically
infeasible to operate any business other than an advertising agency! Civilization will collapse
and any subsequent civilizations will have archaeological sites dedicated to the "marketing era"
strata of the earth's crust.
AFAIK mathematically consistent means that it's all derived from the axioms without any hand waving.
OK, but what are the axioms of which you speak? The brain is an insanely complex physical object and that forces our way of learning about it to be empirical. If we had a formal expression of a brain that could be analyzed mathematically, then the axioms would be cut and dry -- inherent in the formal system.
I guess what I'm saying is that the presence/absence of a formal system is pretty much what defines what is and what isn't mathematics. Though, philosophers of mathematics may argue this point... On the other hand, philosophers can never agree on a point anyway, only dispute it, since philosophy isn't mathematics!
Please correct me if I'm wrong, but I have a basic understanding of homotopy.
I guess you view a solution as a certain kind of map on R^3 that obeys Maxwell's equations and then use homotopy to deform one map into another, all the while respecting Maxwell. Then, one element of the homotopy group would correspond to one family of solutions which may all be transformed one into another via homotopy. Knowledge about the group formed (which has to come from the kind of topological space that Maxwell's equations define) must imply the existence of other families of solutions (e.g., multiplying two known solutions together in the homotopy group to get a new one).
What I'd like to know is how do Maxwell's equations define the topology on the image of the solution maps. Any help?
Sudoku isn't a math puzzle, it's a logic puzzle...
Well, a logic puzzle is `applied logic' (an application of the study of logic). And, applied logic in a formal system is essentially what mathematics is. In fact, (!) the act of solving a soduko puzzle is an instance of a type of graph coloring problem which appears in combinatorics -- hardcore mathematics. Even more interesting, a program which can solve an n^2 x n^2 soduku type puzzle in polynomial time (in n) would be a miraculous thing: it would show that P=NP, since the question of asking whether such an n^2 x n^2 instance is solvable is NP complete! I'm a graduate student in mathematics and this sort of stuff is my cup of tea -- though, I admit that _doing_ Sodoku puzzles doesn't interest me at all. The AMS Notices published a little article about this a while back, but I don't know the citation.
Even so, the initial conditions of a simulation govern all. Since we have no clue where dark matter comes from, how do they input anything reliable concerning the initial distribution of dark matter? From observed distant galaxies still in their infancy? But, I was under the impression that we only had detailed knowledge of dark matter from closer middle aged galaxies. We already know more about these older types of galaxies, which lets us make the inferences that lead to dark matter. Understanding of galactic genesis is still pretty speculative, and is a big part of ongoing research.
So you're saying that gravity is observed to be so weak compared to other forces because it is somehow "diffused" throughout these extra dimensions? If the electro-magentic force is not present in these other dimensions, might that not explain why dark matter is dark?!
Strictly speaking, the "standard model" does not include anything along the lines of WIMPs or other sorts of crazy non-baryonic matter that might be dark matter. Only the "supers symmetric" models claim to have the ability to sort out dark matter. This is the sort of theory that the Large Hadron Collider is supposed to help sort out (in the limit of energy that the standard model _breaks down_). Maybe (?) we'll some answer's after the LHC comes online in a few months.
The article in Wikipedia:
The Physics of Glass
States that the relaxation time for glass is estimated to be on the order of 10^32 years -- that's more than a quadrillion quadrillion years, which also happens to be (what is thought to be) the expected life-span of a proton.
BTW The relaxation time for glass is to be interpreted as "...the average... time for the solid in a metastable supercooled liquid or glass to approach the molecular motion characteristic of a crystal."
-from Relaxation Time
Therefore, any hope of experimentally verifying this hypothesis is off. It's only test would be in the utility of the theory.
... I would hate to think that all that beauty and profundity and goodness could be captured; even approximated in any way at all! in the horribly rigid computational devices of our era.
When you boil it down, humans are just collection carbon, nitrogen, oxygen, and hydrogen (and some other trace elements). What difference does it make if an intelligence is made of mostly "natural" carbon entities vs. mostly "unnatural" silicon entities? I believe this misses Hofstadter's idea. That the "horribly rigid computational devices of our era" are currently implemented with silicon, is immaterial. He means that our minds and consciousnesses are such beautifully complex machines, that the crude computational devices and formal languages we have so far developed are insufficient to model them. Hofstadter shares your sentiment that the medium in which his `strange loops of consciousness' are realized makes no difference at all -- it is the pattern that matters, and only the pattern that matters.
This article in wikipedia explains how all steady state theories were refuted with the discovery of the CMB radiation:
http://en.wikipedia.org/wiki/Steady_State
Apparently, in a steady state universe, there should be a continuous change from the radiation given off by stars/matter and such all the way down to the low energy of the the CMB. This is not observed, and so the CMB radiation must be due to a discrete event which happened `simultaneously' everywhere, i.e., the big bang. At least that's the way I understand it.
Slashdot Mirror
I'm not sure, but isn't it the case that Bose-Einstein condensates have been created (a collection of super cooled particles) and shown to behave as if they were a single particle?
Specifically, haven't physicists achieved putting such a condensate in a superposition? Wouldn't that count as a larger scale demonstration of superposition?
I can see that a virus is going to be much more complex than a bunch of uniform particles, but why should scale matter whether superposition is possible? Don't all objects experience some level of superpostion, only it occurs for immeasurably small time periods before wave collapse?
One question that Schroedinger's Cat raises is "Is there a level of complexity that prevents a superposition forming?" and the proposed experiment will extend the upper limit of complexity that we know quantum mechanics applies to.
Exactly! But, isn't there a different (though related) question: Does it matter if the object (cat) is capable of making observations of its own? Is it possible that "being capable of observation" excludes the possibility of superposition?
I think we'd all agree that a virus isn't capable of making observations... but the cat is? (or if you have a problem with that, just place a person in the box)
Or maybe (BIG maybe), "being capable of observation" just requires a certain amount of complexity?
great theorems of mathematics. This book by William Dunham is a look at the lives and work of 10 or so great mathematicians throughout history.
The sections on Euclid and primes, Euler and infinite sums, and Cantor and the continuum are particularly good. Though it covers great and truly ingenious (even inspiring) mathematics, it should all be accessible to a high school student. Plus, the biographies are interesting in their own right and help to break up the parts where you actually have to concentrate.
I read this in my first year or so as an undergraduate, and I recommend it to anyone who wants to gain a feel for what mathematics is all about.
Exactly, its only the inclusion of gravity within the perspective of special relativity (which is called general relativity) that causes problems with quantum mechanical formulations.
On the contrary, advertising will flourish even more.
It seems that everyone can agree that there is a type of advertising which has as its main function to affect a person's subconscious tendencies toward a particular brand.
Now, if every brand employs this sort of advertising, to a more or less equal degree, then exactly which brand does a person settle on? If the advertising is done right on each brand's behalf, then statistically there will be roughly equal portions of the consumer base that will prefer any particular brand. Any new brand that enters the fray will be at a gross disadvantage unless they employ similar tactics. It will become economically impossible for them to sell their brand name product without a comparative advertising campaign. Also, if any existing brand ups its advertising effectiveness, then the other brands will have to follow suit, or risk losing large portions of the consumer base. So you see, more and better advertising begets more and better advertising, and a vicious cycle ensues.
More advertising agencies will emerge to handle the increasing volume of required advertising. These agencies will compete amongst themselves (through advertising) to be employed by the various brands. Existing agencies will need to increase their output further, too, and so advertisers will begin to pay other advertisers to advertise their advertising!
Eventually, we will reach the advertising event horizon! -- where it becomes economically infeasible to operate any business other than an advertising agency! Civilization will collapse and any subsequent civilizations will have archaeological sites dedicated to the "marketing era" strata of the earth's crust.
AFAIK mathematically consistent means that it's all derived from the axioms without any hand waving.
OK, but what are the axioms of which you speak? The brain is an insanely complex physical object and that forces our way of learning about it to be empirical. If we had a formal expression of a brain that could be analyzed mathematically, then the axioms would be cut and dry -- inherent in the formal system.
I guess what I'm saying is that the presence/absence of a formal system is pretty much what defines what is and what isn't mathematics. Though, philosophers of mathematics may argue this point... On the other hand, philosophers can never agree on a point anyway, only dispute it, since philosophy isn't mathematics!
... it isn't that the escape velocity of a black hole is greater than c, it is that there is no path out of a black hole.
This is correct. Light only follows space-time geodesics.
Please correct me if I'm wrong, but I have a basic understanding of homotopy.
I guess you view a solution as a certain kind of map on R^3 that obeys Maxwell's equations and then use homotopy to deform one map into another, all the while respecting Maxwell. Then, one element of the homotopy group would correspond to one family of solutions which may all be transformed one into another via homotopy. Knowledge about the group formed (which has to come from the kind of topological space that Maxwell's equations define) must imply the existence of other families of solutions (e.g., multiplying two known solutions together in the homotopy group to get a new one).
What I'd like to know is how do Maxwell's equations define the topology on the image of the solution maps. Any help?
Sudoku isn't a math puzzle, it's a logic puzzle...
Well, a logic puzzle is `applied logic' (an application of the study of logic). And, applied logic in a formal system is essentially what mathematics is. In fact, (!) the act of solving a soduko puzzle is an instance of a type of graph coloring problem which appears in combinatorics -- hardcore mathematics. Even more interesting, a program which can solve an n^2 x n^2 soduku type puzzle in polynomial time (in n) would be a miraculous thing: it would show that P=NP, since the question of asking whether such an n^2 x n^2 instance is solvable is NP complete! I'm a graduate student in mathematics and this sort of stuff is my cup of tea -- though, I admit that _doing_ Sodoku puzzles doesn't interest me at all. The AMS Notices published a little article about this a while back, but I don't know the citation.
Yes... I'll call it the ratotron dynamics technique!! the most powerful flan eating creation known to man! Eaaaghaha! --Nova
Even so, the initial conditions of a simulation govern all. Since we have no clue where dark matter comes from, how do they input anything reliable concerning the initial distribution of dark matter? From observed distant galaxies still in their infancy? But, I was under the impression that we only had detailed knowledge of dark matter from closer middle aged galaxies. We already know more about these older types of galaxies, which lets us make the inferences that lead to dark matter. Understanding of galactic genesis is still pretty speculative, and is a big part of ongoing research.
So you're saying that gravity is observed to be so weak compared to other forces because it is somehow "diffused" throughout these extra dimensions? If the electro-magentic force is not present in these other dimensions, might that not explain why dark matter is dark?!
Strictly speaking, the "standard model" does not include anything along the lines of WIMPs or other sorts of crazy non-baryonic matter that might be dark matter. Only the "supers symmetric" models claim to have the ability to sort out dark matter. This is the sort of theory that the Large Hadron Collider is supposed to help sort out (in the limit of energy that the standard model _breaks down_). Maybe (?) we'll some answer's after the LHC comes online in a few months.
The article in Wikipedia: The Physics of Glass States that the relaxation time for glass is estimated to be on the order of 10^32 years -- that's more than a quadrillion quadrillion years, which also happens to be (what is thought to be) the expected life-span of a proton.
BTW The relaxation time for glass is to be interpreted as "...the average ... time for the solid in a metastable supercooled liquid or glass to approach the molecular motion characteristic of a crystal."
-from Relaxation Time
Therefore, any hope of experimentally verifying this hypothesis is off. It's only test would be in the utility of the theory.
This article in wikipedia explains how all steady state theories were refuted with the discovery of the CMB radiation: http://en.wikipedia.org/wiki/Steady_State Apparently, in a steady state universe, there should be a continuous change from the radiation given off by stars/matter and such all the way down to the low energy of the the CMB. This is not observed, and so the CMB radiation must be due to a discrete event which happened `simultaneously' everywhere, i.e., the big bang. At least that's the way I understand it.